1. Sampling Theorem
主講者：虞台文

2. Content Periodic Sampling Sampling of Band-Limited Signals
Aliasing --- Nyquist rate
CFT vs. DTFT
Reconstruction of Band-limited Signals
Discrete-Time Processing of Continuous-Time Signals
Continuous-Time Processing of Discrete-Time Signals
Changing Sampling Rate
Realistic Model for Digital Processing

3. Sampling Theorem
Periodic Sampling

4. Continuous to Discrete-Time Signal Converter
xc(t)
x(n)= xc(nT)
C/D T
Sampling rate

5. Conversion from impulse train to discrete-time sequence
C/D System
Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)

6. Sampling with Periodic Impulse train
xc(t) T 2T 3T 4T T
2T
3T t
xc(t) 2T 4T 8T
10T
2T
4T
8T n
x(n) 1 2 3 4 1 2 3 n
x(n) 2 4 6 8 2 4 6

7. Sampling with Periodic Impulse train
We want to restore xc(t) from x(n).
Sampling with Periodic Impulse train
What condition has to be placed on the sampling rate? t
xc(t) T 2T 3T 4T T
2T
3T n
x(n) 1 2 3 4 1 2 3 8T
10T
4T
8T 6 8 4 6

8. Conversion from impulse train to discrete-time sequence
C/D System
Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)

9. Conversion from impulse train to discrete-time sequence
C/D System
Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)

10. C/D System
s:
Sampling Frequency

11. C/D System

12. Sampling of Band-Limited Signals
Sampling Theorem
Sampling of
Band-Limited Signals

13. Band-Limited Signals Xc(j) Band-Limited  Band-Unlimited Yc(j)  N1
Band-Limited 
Yc(j)
Band-Unlimited

14. Sampling of Band-Limited Signals
Xc(j) N
N 1
Band-Limited
Sampling with
Higher Frequency  s
s
2s
3s
2s
3s
S(j)
2/T
Sampling with
Lower Frequency 
4s
4s
2s
6s
2s
6s
S(j)
2/T

15. Aliasing --- Nyquist Rate
Sampling Theorem
Aliasing ---
Nyquist Rate

16. Recoverability s > 2N s < 2N Xc(j) Band-Limited  S(j) N1 s
s
2s
3s
2s
3s
S(j)
2/T
4s
4s
6s
6s
Sampling with
Higher Frequency
Lower Frequency
Recoverability
s > 2N
s < 2N

17. Case 1: s > 2N Xc(j)  S(j)  Xs(j)  N N 1 s s 2s 3s
2/T  s
s
2s
3s
2s
3s
Xs(j)
1/T

18. Case 1: s > 2N Xs(j) is a periodic function with period s.
Xc(j) N
N 1 s
s
2s
3s
2s
3s
S(j)
2/T
1/T
Xs(j)
Case 1: s > 2N
Passing Xs(j) through a low-pass filter with cutoff frequency N < c< s N , the original signal can be recovered.
Xs(j) is a periodic function with period s.

19. Case 2: s < 2N Xc(j)  S(j)  Xs(j)  N N 1 2s 2s 4s
2/T 
2s
2s
4s
6s
4s
6s
Xs(j)
1/T

20. Aliasing Case 2: s < 2N No way to recover the original signal.
Xc(j) N
N 1
1/T
2s
2s
4s
6s
4s
6s
S(j)
2/T
Xs(j)
Case 2: s < 2N
Xs(j) is a periodic function with period s.
No way to recover the original signal.
Aliasing

21. Nequist Rate Xc(j) Band-Limited  Nequist frequency (N)1
Band-Limited
Nequist frequency (N)
The highest frequency of a band-limited signal
Nequist rate = 2N

22. Nequist Sampling Theorem
Xc(j) N
N 1
Band-Limited
s > 2N
Recoverable
s < 2N
Aliasing

23. Sampling Theorem
CFT vs. DTFT

24. Conversion from impulse train to discrete-time sequence
C/D System
Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)

25. Continuous-Time Fourier Transform
Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)

26. Conversion from impulse train to discrete-time sequence
CFT vs. DTFT
Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)
x(n)

27. Conversion from impulse train to discrete-time sequence
xc(t)
x(n)= xc(nT) 
s(t)
xs(t)
x(n)
CFT vs. DTFT

28. CFT vs. DTFT

29. CFT vs. DTFT 
Xc(j) 1 
Xs(j) s
s
1/T 
X(ej) 2
1/T
2 4
4

30. CFT vs. DTFT s2 Xc(j)  Xs(j)  X(ej)  Amplitude scaling1
Amplitude scaling
&
Repeating 
Xs(j) s
s
1/T
Frequency scaling
s2 
X(ej) 2
1/T
2 4
4

31. Reconstruction of Band-limited Signals
Sampling Theorem
Reconstruction of Band-limited Signals

32. Key Concepts t  Xc(j) CFT ICFT Sampling C/D n DTFT  X(ej)  
xc(t) T 2T 3T 4T T
2T
3T 
Xc(j)
/T
/T
CFT
ICFT
Sampling
C/D
Retrieve
One period n
x(n) 1 2 3 4 1 2 3
DTFT 
X(ej)  
IDTFT

33. Interpolation

34. Interpolation
n(t)
x(n)

35. Ideal D/C Reconstruction System
x(n)
xs(t)
xr(t)
Covert from sequence to impulse train T
Ideal Reconstruction Filter
Hr(j)

36. Ideal D/C Reconstruction System
Obtained from sampling xc(t) using an ideal C/D system.
Ideal D/C Reconstruction System
x(n)
xs(t)
xr(t)
Covert from sequence to impulse train T
Ideal Reconstruction Filter
Hr(j) 
/T
/T
Hr(j) T

37. Ideal D/C Reconstruction System
x(n)
xs(t)
xr(t)
Covert from sequence to impulse train T
Ideal Reconstruction Filter
Hr(j)

38. Ideal D/C Reconstruction System
x(n)
xs(t)
xr(t)
Covert from sequence to impulse train T
Ideal Reconstruction Filter
Hr(j)
D/C

39. Ideal D/C Reconstruction System
xc(t)
C/D T
x(n)
xr(t)
D/C T
In what condition xr(t) = xc(t)?

40. Discrete-Time Processing of Continuous-Time Signals
Sampling Theorem
Discrete-Time Processing of Continuous-Time Signals

41. The Model xc(t) yr(t) xc(t) x(n) y(n) yr(t) T C/D D/C T
Discrete-Time
System
Continuous-Time
System
xc(t)
yr(t)

42. The Model Heff(j) H (ej) y(n) yr(t) xc(t) x(n) D/C T C/D
Discrete-Time
System
xc(t)
C/D
x(n)
Continuous-Time
H (ej)
Heff(j)

43. LTI Discrete-Time Systems
xc(t)
C/D
x(n)
y(n)
yr(t)
H (ej)
Discrete-Time
System
D/C T

44. LTI Discrete-Time Systems
y(n)
yr(t)
D/C T
Discrete-Time
System
xc(t)
C/D
x(n)
H (ej)

45. LTI Discrete-Time Systems
Continuous-Time
System
xc(t)
yr(t)
Heff(j)

46. Example:Ideal Lowpass Filter
y(n)
yr(t)
D/C T
Discrete-Time
System
xc(t)
C/D
x(n) 1  c
c
H(ej)

47. Example:Ideal Lowpass Filter1  c
c
Heff(j)
Continuous-Time
System
xc(t)
yr(t)

48. Example: Ideal Bandlimited Differentiator
Continuous-Time
System
xc(t)

49. Example: Ideal Bandlimited Differentiator
|Heff(j)|
Continuous-Time
System
xc(t)

50. Example: Ideal Bandlimited Differentiator
Continuous-Time
System
xc(t) 
|Heff(j)|

51. Impulse Invariance xc(t) yc(t)
Continuous-Time
LTI system
hc(t), Hc(j)
xc(t)
yc(t)
y(n)
yc(t)
D/C T
Discrete-Time
LTI System
h(n)
H(ej)
xc(t)
C/D
x(n)
What is the relation between hc(t) and h(n)?

52. Impulse Invariance

53. Impulse Invariance

54. Impulse Invariance xc(t) yc(t)
Continuous-Time
LTI system
hc(t), Hc(j)
xc(t)
yc(t)
y(n)
D/C T
Discrete-Time
LTI System
h(n)
H(ej)
C/D
x(n)
What is the relation between hc(t) and h(n)?

55. Continuous-Time Processing of Discrete-Time Signals
Sampling Theorem
Continuous-Time Processing of Discrete-Time Signals

56. The Model x(n) y(n) yc(t) y(n) x(n) xc(t) C/D T D/C Discrete-Time
Continous-Time
System
x(n)
D/C
xc(t)
Discrete-Time
System
x(n)
y(n)

57. The Model H (ej) Hc(j) yc(t) y(n) x(n) xc(t) C/D T D/C Discrete-Time
Continous-Time
System
x(n)
D/C
xc(t)
Discrete-Time
Hc(j)
H (ej)

58. The Model
yc(t)
y(n)
C/D T
Continous-Time
System
x(n)
D/C
xc(t)
Hc(j)

59. The Model

60. The Model
Discrete-Time
System
x(n)
y(n)
H (ej)

61. The Model H (ej) Hc(j) x(n) y(n) yc(t) y(n) x(n) xc(t) C/D T D/C
Continous-Time
System
x(n)
D/C
xc(t)
Hc(j)
Discrete-Time
System
x(n)
y(n)
H (ej)

62. Changing Sampling Rate Using Discrete-Time Processing
Sampling Theorem
Changing Sampling Rate Using
Discrete-Time Processing

63. The Goal
Down/Up
Sampling

64. Sampling Rate Reduction By an Integer Factor
Down/Up
Sampling
Down
Sampling

65. Sampling Rate Reduction By an Integer Factor

66. Sampling Rate Reduction By an Integer Factor
Let r = kM + i

67. Sampling Rate Reduction By an Integer Factor
Xc(j)  N
N
Xs(j), X (ejT) 
2/T
2/T
1/T
N=NT
N
X (ej)  2
2
1/T

68. Sampling Rate Reduction By an Integer Factor
Xc(j) 
Xs(j), X (ejT)
2/T
2/T
1/T
N=NT
N
X (ej)  2
2
N <  : no aliasing
Sampling Rate Reduction By an Integer Factor
M=2
Xd (ej)  2
2
1/MT
Xd (ejT) 
1/T’
2/T’
2/T’
4/T’
4/T’

69. Antialiasing Aliasing M=3 N N X (ej)  2 2 1/T  Xd (ej) 1/MT

70. Antialiasing However, xd(n) x(nT’) M=3 N N X (ej)  2 2 1/T
/3
Hd (ej)  2
2 1
/3
M=3  2
2
/3
/3  2
2

71. Decimator
Lowpass filter
Gain = 1
Cutoff = /M M

72. Increasing Sampling Rate By an Integer FactorUp
Sampling T

73. Increasing Sampling Rate By an Integer FactorUp
Sampling
X (ej)   
1/T
X’ (ej)   
L/T

74. Interpolator
Lowpass filter
Gain = L
Cutoff = /L L

75. Interpolator

76. Interpolator L=3 X (ej)    1/T Xe(ej)    1/T   Hi(ej) 
/3
/3
Xi(ej)   
L/T

77. Changing the Sampling Rate By a Noninteger Factor
Resampling

78. Changing the Sampling Rate By a Noninteger Factor
Sampling Periods:
Lowpass filter
Gain = 1
Cutoff = /M M
Gain = L
Cutoff = /L L M
Lowpass filter
Gain = L
Cutoff = min(/L, /M) L

79. Realistic Model for Digital Processing
Sampling Theorem
Realistic Model for
Digital Processing

80. Ideal Discrete-Time Signal Processing Model
y(n)
yc(t)
D/C T
Discrete-Time
LTI System
xc(t)
C/D
x(n)
Real world signal usually is not bandlimited
Ideal continuous-to-discrete converter is not realizable
Ideal discrete-to-continuous converter is not realizable

81. Compensated reconstruction filter
More Realistic Model
y(n)
yc(t)
D/C T
Discrete-Time
LTI System
xc(t)
C/D
x(n)
Anti-aliasing filter
Sample and Hold
A/D converter
Discrete-time system
D/A converter
Compensated reconstruction filter T

82. Analog-to-Digital Conversion
Sample and
Hold
A/D converter T T

83. Sample and Hold t T
ho(t) t T

84. Sample and Hold t
xo(t) t T
ho(t) T

85. Sample and Hold t T
ho(t)
xo(t)

86. Sample and Hold Goal: To hold constant sample value for A/D converter.
Zero-Order
Hold
ho(t)

87. A/D Converter
C/D T
Quantizer
Coder

88. Typical Quantizer 2’s complement code Offset binary 011 010 001 000
111
110
101
100
2Xm
(B+1)-bit Binary code

89. Analysis of Quantization Errors
C/D T
Quantizer
Coder
Quantizer
Q[ ]

90. Analysis of Quantization Errors
The error sequence e(n) is a stationary random process.
e(n) and x(n) are uncorrelated.
The random variables of the error process are uncorrelated, i.e., the error is a white-noise process.
e(n) is uniform distributed.

91. SNR (Signal-to-Noise Ratio)

92. SNR (Signal-to-Noise Ratio)
每增加一個bit，SNR增加約6dB

93. SNR (Signal-to-Noise Ratio)
Let x=Xm / 4  SNR  6B1.25 dB
SNR (Signal-to-Noise Ratio)
x大較有利，但不得過大(為何？)
x過小不利 x每降低一倍SNR少6dB
X~N(0, x2)  P(|X|<4x )